1. At the end of each year, a credit rating company ranks the performance of corporate bonds in particular sector and classifies them into (simplified) four credit ratings A, B, C, and D (with rating A being the best/top performance). From past credit rating history, a bond will either remain in the same rating with probability 1/2 or will get downgraded one level down or will get upgraded one level up with the same probability. Furthermore, it has been found that a bond that has remained in the top or bottom rat- ing for two consecutive years has a probability 5/6 of remaining in the same rating the following year. By subdividing the top and bottom rating into A+ and D-, the changes in the credit rating may be described by a time homogeneous discrete-time Markov chain X = {X,,n = N} with the one-year (step) transition probability matrix A+ A B C D D- A+ (5/6 0 1/6 0 0 0 P = ABC 1/2 0 1/2 0 0 0 0 1/4 1/2 1/4 0 0 (1) 0 0 1/4 1/2 1/4 0 D 0 0 0 1/2 0 1/2 D- 0 0 0 1/6 0 5/6 (a) Write down the state space S of the Markov chain. (b) Draw the state diagram of the Markov chain with indicated arrow and probability. (c) Explain whether the Markov chain is irreducible, aperiodic and positive recurrent. (d) Using the Bayes formula P(X2 = A|X₁ = B) = P(X₂ = A, X₁ = x[X0 = B), XES what is the probability that B rated bond will get updated to rating A in two years? (e) Find stationary distribution л of X as the unique solution of лP = π.
1. At the end of each year, a credit rating company ranks the performance of corporate bonds in particular sector and classifies them into (simplified) four credit ratings A, B, C, and D (with rating A being the best/top performance). From past credit rating history, a bond will either remain in the same rating with probability 1/2 or will get downgraded one level down or will get upgraded one level up with the same probability. Furthermore, it has been found that a bond that has remained in the top or bottom rat- ing for two consecutive years has a probability 5/6 of remaining in the same rating the following year. By subdividing the top and bottom rating into A+ and D-, the changes in the credit rating may be described by a time homogeneous discrete-time Markov chain X = {X,,n = N} with the one-year (step) transition probability matrix A+ A B C D D- A+ (5/6 0 1/6 0 0 0 P = ABC 1/2 0 1/2 0 0 0 0 1/4 1/2 1/4 0 0 (1) 0 0 1/4 1/2 1/4 0 D 0 0 0 1/2 0 1/2 D- 0 0 0 1/6 0 5/6 (a) Write down the state space S of the Markov chain. (b) Draw the state diagram of the Markov chain with indicated arrow and probability. (c) Explain whether the Markov chain is irreducible, aperiodic and positive recurrent. (d) Using the Bayes formula P(X2 = A|X₁ = B) = P(X₂ = A, X₁ = x[X0 = B), XES what is the probability that B rated bond will get updated to rating A in two years? (e) Find stationary distribution л of X as the unique solution of лP = π.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:1. At the end of each year, a credit rating company ranks the performance of corporate bonds
in particular sector and classifies them into (simplified) four credit ratings A, B, C, and
D (with rating A being the best/top performance). From past credit rating history, a bond
will either remain in the same rating with probability 1/2 or will get downgraded one
level down or will get upgraded one level up with the same probability.
Furthermore, it has been found that a bond that has remained in the top or bottom rat-
ing for two consecutive years has a probability 5/6 of remaining in the same rating the
following year.
By subdividing the top and bottom rating into A+ and D-, the changes in the credit rating
may be described by a time homogeneous discrete-time Markov chain X = {X,,n = N}
with the one-year (step) transition probability matrix
A+ A
B
C
D
D-
A+ (5/6 0
1/6
0
0
0
P =
ABC
1/2
0
1/2
0
0
0
0
1/4
1/2
1/4
0
0
(1)
0
0
1/4 1/2
1/4
0
D
0
0
0
1/2 0
1/2
D-
0
0
0
1/6
0
5/6
(a) Write down the state space S of the Markov chain.
(b) Draw the state diagram of the Markov chain with indicated arrow and probability.
(c) Explain whether the Markov chain is irreducible, aperiodic and positive recurrent.
(d) Using the Bayes formula
P(X2 = A|X₁ = B) = P(X₂ = A, X₁ = x[X0 = B),
XES
what is the probability that B rated bond will get updated to rating A in two years?
(e) Find stationary distribution л of X as the unique solution of лP = π.
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